Strong electron-phonon coupling in the rare-earth carbide superconductor La$_2$C$_3$

Superconductivity in rare earth carbides has attracted interested again after the recent discovery of the 18 K superconductor Y$_2$C$_3$. Here, we present the crystal structure as well as the superconducting properties of the rare-earth sesquicarbides La$_2$C$_3$ ($T_c$ $\approx$ 13.4 K) gained from low-temperature neutron powder diffraction, specific heat and electrical resistivity measurements. From a detailed analysis of specific heat as well as the comparison with the full potential electronic structure calculations, a quantitative estimate of the electron-phonon coupling strength and the logarithmic average phonon frequency is made. The electron-phonon coupling constant found to be $\lambda_{ph}$ $\sim$ 1.35, and the low energy phonon modes are responsible for the superconductivity. These results suggest that La$_2$C$_3$ is in the strong coupling regime and the relevant phonon modes are the La-related modes rather than the C-C stretching modes. The upper critical fields($H_{c2}$) show a clear enhancement with respect to the Werthamer-Helfand-Hohenberg prediction and amount to $H_{c2}(0)$ $\sim$ 20 T confirming the strong electron-phonon coupling.